Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories

We present an analysis of Brownian force noise from residual gas damping of reference test masses as a fundamental sensitivity limit in small force experiments. The resulting acceleration noise increases significantly when the distance of the test mass to the surrounding experimental apparatus is smaller than the dimension of the test mass itself. For the Advanced LIGO interferometric gravitational wave observatory, where the relevant test mass is a suspended 340 mm diameter cylindrical end mirror, the force noise power is increased by roughly a factor 40 by the presence of a similarly shaped reaction mass at a nominal separation of 5 mm. The force noise, of order $20\mathrm{fN}/{\mathrm{Hz}}^{1/2}$ for $2\times {10}^{-6}\mathrm{Pa}$ of residual ${\mathrm{H}}_2$ gas, rivals quantum optical fluctuations as the dominant noise source between 10 and 30 Hz. We present here a numerical and analytical analysis for the gas damping force noise for Advanced LIGO, backed up by experimental evidence from several recent measurements. Finally, we discuss the impact of residual gas damping on the gravitational wave sensitivity and possible mitigation strategies.

Figure 1

Cartoon illustration of the Advanced LIGO experimental configuration studied and simulated here, seen perpendicular to the symmetry axes of the cylindrical test (TM) and reaction (RM) masses, with radii $R=17\mathrm{cm}$ and heights $h=20\mathrm{cm}$ and $h_{\mathrm{RM}}=13\mathrm{cm}$. The gap $d$, nominally 5 mm for Advanced LIGO, is varied in the simulations. The simulated experimental enclosure, not shown to scale here, is a coaxial cylinder of radius $R_S=10R$ and length $H_S=11h+h_{\mathrm{RM}}+d$.

Figure 2

Simulation data for the scatter in the total momentum $\Delta q$ transferred to the TM in simulation time $T_0$, for several values of the intercylinder gap $d$, each fit to the model in Eq. (6). The range of gaps studied represents aspect ratios ($2R/D$) from nearly 7 to 340. The dashed gray line represents the prediction in the limit of infinite gap, $S_F^{\infty }$, which is recovered for very short simulation times. Data are scaled for a reference pressure $p=2\times {10}^{-6}\mathrm{Pa}$.

Figure 3

Simulation data (dots) for the force noise linear spectral density as a function of the gap, $d$, for $p=2\times {10}^{-6}\mathrm{Pa}$, with a fit to the simple analytical model [Eq. (20)]. The dashed gray line is the theoretical infinite gap limit. Points at $d$ of 5 and 20 mm are circled to indicate, respectively, the geometries of the baseline and alternative designs discussed in the conclusion. Points shown as black “$\times$” are simulation data for the molecular escape time, converted into force noise using the model in Sec. 3a.

Figure 4

Value of the characteristic time constant $\tau$ extracted from the simulation force fluctuations (dots), as a function of gap $d$, using the model in Eq. (6). The dashed curve represents the raw result of the approximate analytic model [Eq. (14)] in Sec. 3b, while the solid curve represents a two parameter fit to the same model, discussed in the text. Again, circled points at 5 and 20 mm indicate the baseline and alternate geometries. Points marked with a black “$\times$” are simulation values for the average molecular escape time from the intercylinder gap, which correspond approximately to the values of $\tau$ extracted from the force fluctuations.

Figure 5

Simulation force noise linear spectral density as a function of frequency for 5 and 20 mm gaps, using $p=2\times {10}^{-6}\mathrm{Pa}$. Also plotted, with solid black lines, is the model from Eq. (5), using the parameters extracted from the data in Fig. 2. The dashed gray line corresponds to the force noise in the infinite volume limit.

Figure 6

Plot of experimental data from the UTN gas-damping experiment [5] featuring cubic test masses in two different torsion pendulums dominated by rotational (1TM) and translational (4TM) squeezing, the latter with two different conditions of proximity to the surrounding apparatus. The pressure independent damping has been removed from each data set to show only the gas contribution. Linear fits to each data set are shown. The simulation predictions for the three cases are shown as thick black lines, while the infinite volume limit predictions for the two pendulums are shown in dashed gray.

Figure 7

Plot of experimental data from the UW gas-damping experiment [6] for two different torsion pendulums, labeled NTA and LTA, both with a rectangular plate geometry in proximity of a second parallel plate at distance $d$. The pressure independent damping has been removed from each data set to show only the gas contribution. Simulation points are shown with black $\times$, with infinite volume limits shown in dashed gray. Dotted lines connecting simulation points are included as a guide for the eye.

Figure 8

Gravitational wave strain measurement noise contributions for Advanced LIGO [19]. Dashed and solid curves correspond to the noise contributions from gas-damping noise with $2\times {10}^{-6}\mathrm{Pa}$ of ${\mathrm{H}}_2$ for the nominal geometry (all 5 mm gaps between all four TM–reaction masses pairs) and the alternate geometry with a 20 mm gap adjacent to the two input TM. The total measurement noise in the two geometries are shown with thick gray curves.

Figure 9

Illustration of the geometry used for calculating the mean square lateral displacement and the mean time of flight for a molecule emitting into the gap from the center of one of the cylinders. The angle ${\theta }_C$ represents the critical value of the polar angle $\theta$, beyond which emission from the center results in the molecule escaping the gap laterally in a single flight.